a method for multiple attribute decision making based on the fusion of multisource information

Research ArticleA Method for Multiple Attribute Decision Making Based on the Fusion of Multisource Information F.. Recently, multiple attribute decision making MADM problems [1,2], whose

Research Article

A Method for Multiple Attribute Decision Making Based on

the Fusion of Multisource Information

F W Zhang,1,2S H Xu,3B J Wang,1,2and Z J Wu1,2

1 Jiangsu Key Laboratory of Urban ITS, Southeast University, Nanjing 210096, China

2 Jiangsu Province Collaborative Innovation Center of Modern Urban Traffic Technologies, Nanjing 210096, China

3 Department of Mathematics, Zhaoqing University, Zhaoqing 526061, China

Correspondence should be addressed to F W Zhang; fangweizhang80@yahoo.com.cn

Received 29 October 2013; Accepted 21 January 2014; Published 3 March 2014

Academic Editor: Ljubisa Kocinac

Copyright © 2014 F W Zhang et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

We propose a new method for the multiple attribute decision making problem In this problem, the decision making information assembles multiple source data Two main advantages of this proposed approach are that (i) it provides a data fusion technique, which can efficiently deal with the multisource decision making information; (ii) it can produce the degree of credibility of the entire decision making The proposed method performs very well especially for the scenario that there exists conflict among the multiple source information Finally, a traffic engineering example is given to illustrate the effect of our method

1 Introduction

In the decision-making theory, many methods and their

applications have been extensively studied Recently, multiple

attribute decision making (MADM) problems [1,2], whose

decision making information comes from multiple source

data, receive more and more attention Among these

prob-lems, the MADM problems which have the subjective and the

objective information [1–5] at the same time, and the multiple

attribute group decision making (MAGDM) [6–9] problems

are the two hot topics in this research field

The key to the two kinds of problems is to fuse various

pieces of information [10] For example, the following

liter-ature is to solve the first kind of problems The literliter-ature [3]

has proposed an optimization model to deal with the MADM

problems with preference information on alternatives, which

were given by decision maker in a fuzzy relation With respect

to the MADM problems with intuitionists fuzzy information,

the literature [4] has proposed an optimization model based

on the maximum deviation method By this model, we can

derive a simple and an exact formula for determining the

completely unknown attribute weights The literature [5] has

proposed a linguistic weighted arithmetic averaging operator

to solve the MADM problems, where there is linguistic

preference information and the preference values take the form of linguistic variables and so forth

In the respect of MAGDM problems, the literature [6] has researched the MAGDM problem with different formats

of preference information on attributes; the literature [7] has researched the 2-tuple linguistic MAGDM problems with incomplete weight information and established an optimiza-tion model based on the maximizing deviaoptimiza-tion method; the literature [8] has presented a new approach to the MAGDM problems, where cooperation degree and reliability degree are proposed for aggregating the vague experts’ opinions; the literature [9] has developed a compromise ratio methodology for fuzzy MAGDM problems and so forth

Through these literatures, we could find that most of the solutions have used some subjective attitudes or information [10,11], which were not provided by the problem itself This

is seriously out of line with the social needs In order to overcome this defect, this paper presents two methods for the above two kinds of problems The proposed methods are based on strong calculation and combined with the optimization theory [12] or the variation coefficient method [13]

The highlights of this new method could be summarized into two points The first, it can efficiently deal with the

http://dx.doi.org/10.1155/2014/972159

multisource decision making information; the second, it

could provide the credibility degree of the final decisional

results

The rest parts of this paper will be organized as follows

In Section 2, we introduce the problems which the article

would explore; in Section3, we introduce the main tool of

our research; in Section4, we put forward two new decision

methods; in Section5, an application example is presented

to illustrate the new method; in Section6, we make some

conclusions and present some further studies

2 Two Problems

2.1 The MADM Problems under the Condition of Information

Conflict We will introduce this problem as follows Let𝑋 =

{𝑥1, 𝑥2, , 𝑥𝑚} be a discrete set of 𝑚 feasible alternatives, let

𝐹 = {𝑓1, 𝑓2, , 𝑓𝑛} be a finite set of attributes, and let 𝑦𝑖𝑗 =

𝑓𝑗(𝑥𝑖) (𝑖 = 1, 2, , 𝑚; 𝑗 = 1, 2, , 𝑛) be the values of the

alternative𝑥𝑖under the attribute𝑓𝑗 In this paper, we only

consider the situation that𝑦𝑖𝑗is given in real numbers The

decision matrix of attribute set𝐹 with regard to the set 𝑋 is

expressed by the matrix

𝑌 = (

𝑦11 𝑦12 ⋅ ⋅ ⋅ 𝑦1𝑛

𝑦21 𝑦22 ⋅ ⋅ ⋅ 𝑦2𝑛

. d .

𝑦𝑚1 𝑦𝑚2 ⋅ ⋅ ⋅ 𝑦𝑚𝑛

For convenience, we suppose that the decision matrix𝑌

has been normalized and denote𝑀 = {1, 2, , 𝑚}, 𝑁 =

{1, 2, , 𝑛} For specific details of standardization, please see

the literature [3,14]

The experts have provided the subjective preference

infor-mation for the alternative set𝑋 We denote the information

as𝜆 = (𝜆1, 𝜆2, , 𝜆𝑚)𝑇, in which𝜆𝑖∈ [0, 1], 𝑖 ∈ 𝑀

Based on the above conditions, the problem is to select

and rank the alternatives In this paper, we mainly consider

the situation where there are serious conflicts between the

subjective information and the objective information [15]

2.2 The MAGDM Problems with Interval Vectors In this

subsection, we will introduce a kind of MAGDM problems

with interval vectors The basic concepts are the same as the

above subsection, and we use the mathematical symbols, such

as𝑋 = {𝑥1, x2, , 𝑥𝑚}, 𝐹 = {𝑓1, 𝑓2, , 𝑓𝑛}, 𝑦𝑖𝑗 = 𝑓𝑗(𝑥𝑖),

𝑊 = (𝑤1, 𝑤2, , 𝑤𝑛)𝑇, and the matrix𝑌 directly

Here, the same as the above subsection, we suppose that

the decision matrix 𝑌 has been normalized, and we only

consider the situation that𝑦𝑖𝑗is given in real numbers

Unlike the above subsection, here, the experts do not

pro-vide the subjective preference information for the alternative

set𝑋 but provide the weight information directly Consider

𝐷 = {𝑑1, 𝑑2, , 𝑑𝑡} as the collection of experts, and denote

the weight vectors which are provided by𝐷 as

𝑊1= ([𝑎11, 𝑏11], [𝑎12, 𝑏12] , , [𝑎1𝑛, 𝑏1𝑛])𝑇,

𝑊2= ([𝑎21, 𝑏21], [𝑎22, 𝑏22] , , [𝑎2𝑛, 𝑏2𝑛])𝑇,

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

𝑊𝑡= ([𝑎𝑡1, 𝑏𝑡1], [𝑎𝑡2, 𝑏𝑡2] , , [𝑎𝑡𝑛, 𝑏𝑡𝑛])𝑇

(2) Here,0 ≤ 𝑎𝑘𝑗≤ 𝑏𝑘𝑗≤ 1, 𝑘 = 1, 2, , 𝑡, 𝑗 = 1, 2, , 𝑛 The problem is to solve the MAGDM problem with the above conditions

3 Main Tool of Our Research

The common character of the two problems is that they all involve the operation of interval numbers In addition,

we must point out that the situation we have no weight information equals to the situation where the weight is a variable located in the interval[0, 1] In the following, we would give a new method for operating the interval numbers The new method originates from the basic of strong calculation by modern computer

Without loss of generality, we take calculating the distance between ([𝑎11, 𝑏11], [𝑎12, 𝑏12], , [𝑎1𝑛, 𝑏1𝑛]) and ([𝑎21, 𝑏21], [𝑎22, 𝑏22], , [𝑎2𝑛, 𝑏2𝑛]) as example The detailed procedure is illustrated as follows

Step 1 Divide each[𝑎𝑝𝑞, 𝑏𝑝𝑞] (𝑝∈ {1, 2}, 𝑞 ∈ {1, 2, , 𝑛}) into𝑛∗ parts The value𝑛∗ depends on the demand of the decision makers Then, we will get a set of segmentation points as

̃𝑆 = {𝑎𝑝𝑞, 𝑎𝑝𝑞+𝑛1∗(𝑏𝑝𝑞− 𝑎𝑝𝑞) , 𝑎𝑝𝑞 +2

𝑛∗(𝑏𝑝𝑞− 𝑎𝑝𝑞) , , 𝑏𝑝𝑞}

(3)

We represent each interval [𝑎𝑝𝑞, 𝑏𝑝𝑞] (𝑝∈ {1, 2},

𝑞 ∈ {1, 2, , 𝑛}) by ̃𝑆 Then, we represent the two vectors ([𝑎11, 𝑏11], [𝑎12, 𝑏12], , [𝑎1𝑛, 𝑏1𝑛]) and ([𝑎21, 𝑏21], [𝑎22, 𝑏22], , [𝑎2𝑛, 𝑏2𝑛]) by two sets of real-valued vectors Denote the two sets as𝑃 and 𝑄

Step 2 Take any element𝑝 from 𝑃 and take any element 𝑞 from𝑄; according to the formula

󵄨󵄨󵄨󵄨

󵄨𝑊𝑖− 𝑊𝑗󵄨󵄨󵄨󵄨󵄨

= √(𝑤𝑖1− 𝑤𝑗1)2+ (𝑤𝑖2− 𝑤𝑗2)2+ ⋅ ⋅ ⋅ + (𝑤𝑖𝑛− 𝑤𝑗𝑛)2,

(4)

we could calculate the distance By doing so, we could get (𝑛∗)2𝑛distances Then, we calculate the average value of these distances and denote the average value as𝑑

Step 3 Increase the value𝑛∗gradually and repeat the above steps When the value𝑑 holds steady to two digits after the decimal point, end the procedure and see the final result𝑑∗as the distance between([𝑎11, 𝑏11], [𝑎12, 𝑏12], , [𝑎1𝑛, 𝑏1𝑛]) and ([𝑎21, 𝑏21], [𝑎22, 𝑏22], , [𝑎2𝑛, 𝑏2𝑛])

Obviously, the main advantage of this method is that the calculation procedure is in an objective, consistent way, and

there is no subjective information involved in the calculation

procedure

4 Decision Methods

At the beginning of this section, we would introduce a

method called the simple additive weighting method [1]

Now, we consider a problem in hypothetical situation, where

we have known the weight vector𝑊 = (𝑤1, 𝑤2, , 𝑤𝑛)𝑇

and the attribute values In this situation, we could get the

comprehensive attribute value𝑍𝑖(𝑖 = 1, 2, , 𝑚) by

𝑍𝑖(𝑊) =∑𝑛

𝑗=1

𝑤𝑗𝑦𝑖𝑗 (𝑖 ∈ 𝑀, 𝑗 ∈ 𝑁) (5) Obviously, the bigger𝑍𝑖(𝑊) leads to the more excellent

𝑥𝑖 Therefore, we could accomplish the process of getting the

best alternative and ranking all of the alternatives by (5),

and we could see that the determination of the weight vector

𝑊 = (𝑤1, 𝑤2, , 𝑤𝑛)𝑇is the core of the MADM problem in

general conditions

4.1 Decision Method 1 In this paper, we only consider the

situation where there is significant difference among these

weight vectors; that is, the Kendall consistence [13] of the

above weight vectors is imperfect

The following, we present a new decision making method

for solving the problems of subsection2.1 The characteristic

of our method is that it could provide the credibility of the

decision maker for the subjective information as well as the

entire decisional results Specific decision steps are as follows

Step 1 Solve the single objective programming model

max 𝑍𝑖=∑𝑛

𝑗=1

𝑤𝑗𝑦𝑖𝑗,

s.t 𝑤𝑗 ≥ 0, 𝑗 ∈ 𝑁, ∑𝑛

𝑗=1

𝑤𝑗= 1,

(6)

and denote the result of model (6) as 𝑍max

𝑍max

𝑖 is the ideal value of the comprehensive attribute value

of𝑥𝑖 (𝑖 ∈ 𝑀)

Solve the single objective programming model

min 𝑍𝑖=∑𝑛

𝑗=1

𝑤𝑗𝑦𝑖𝑗,

s.t 𝑤𝑗≥ 0, 𝑗 ∈ 𝑁, ∑𝑛

𝑗=1

𝑤𝑗= 1,

(7)

and denote the result of model (7) as𝑍min𝑖 (𝑖 ∈ 𝑀), and

𝑍min𝑖 is the negative ideal value of the comprehensive attribute

value of𝑥𝑖 (𝑖 ∈ 𝑀)

Step 2 Denote

𝜆∗𝑖 = 𝑍𝑖− 𝑍𝑖min

𝑍max

𝑖 − 𝑍min 𝑖

and establish one single objective optimal model min 󵄨󵄨󵄨󵄨𝜆∗

− 𝜆󵄨󵄨󵄨󵄨2=∑𝑚 𝑖=1󵄨󵄨󵄨󵄨𝜆∗

𝑖 − 𝜆𝑖󵄨󵄨󵄨󵄨2,

s.t 𝑤𝑗 ≥ 0, 𝑖 ∈ 𝑀, 𝑗 ∈ 𝑁, ∑𝑛

𝑗=1

𝑤𝑗= 1

(9)

Step 3 Solve the model (9), and we would get the weight vector𝑊∗ = (𝑤∗1, 𝑤∗2, , 𝑤∗𝑛)𝑇 Up to this point, we could calculate the comprehensive attribute values of each𝑥𝑖 (𝑖 ∈ 𝑀) by (5) Then, we could rank the alternatives and get the optimal alternative𝑥∗

Step 4 If the optimal solution 𝑥∗ is consistent with the subjective decision information𝜆, we consider it as the final optimal solution of the entire decision making process, and consider

𝜂1= 1 − √(𝑤∗

1 −1𝑛)2+ (𝑤∗

2−𝑛1)2+ ⋅ ⋅ ⋅ + (𝑤∗

𝑛−1𝑛)2

⋅ (√𝑛 − 1𝑛 )

−1

(10)

as the degree of believing for the subjective information

In (10), the value √(𝑛 − 1)/𝑛, which is obtained by optimization theory is the max Euclidean distance between ((1/𝑛), (1/𝑛), , (1/𝑛))𝑇 and any possible weight vector

𝑊∗ The value 𝜂1 reflects the similarity scale of 𝑊∗ and ((1/𝑛), (1/𝑛), , (1/𝑛))𝑇 From the aspect of set-valued statistics, the bigger the𝜂1is, the more support would be got from the data of the objective information

Because there is coordination between the subjective and objective information, and they all support the optimal alternative𝑥∗, we set the credibility of the entire decision as 1

Step 5 If the optimal solution 𝑥∗ is inconsistent with the subjective decision information𝜆, we would believe that the subjective information has got no support from the objective information Here, we correct the value𝜂1and set𝜂1as zero Define

̃𝜆𝑝= max {𝜆1, 𝜆2, , 𝜆𝑚} , (11) and define ̃𝑥 as the alternative corresponding to the index

̃𝜆𝑝 Obviously, the alternativẽ𝑥 could represent the subjective information to some extent

Step 6 We use the parameter𝜂2to represent the credibility

of the entire decisional results In this step, we assume that the MADM problems have the alternative set of{𝑥∗, ̃𝑥} only Because the weight information is unknown, we consider the weight vector as random element in weight space

𝑉 = [0, 1] × [0, 1] × ⋅ ⋅ ⋅ × [0, 1] , (12) and the random element follows a uniform distribution

Step 7 By using the main tool of our research, which has

been introduced in Section3, we compare the advantages

of the alternative𝑥∗with the alternative ̃𝑥 and calculate the

credibility of them By (5), every element of𝑉 would support

one optimal alternative Based on this, each alternative would

be supported by a region of hypercube𝑉, and the ranking of

all alternatives could be solved by comparing the regions of

hypercube𝑉 The result of this regions comparison could be

got by the technique of numerical simulation [16]

It’s worth mentioning that if the sum of the weight vector

is not one, by normalization, it is equivalent to a weight vector

with the sum one

4.2 Decision Method 2 Now, we present a new decision

mak-ing method for solvmak-ing the problem of subsection2.2 In this

paper, we only consider the situation where there is significant

difference among these weight vectors𝑊1, 𝑊2, , 𝑊𝑡; that

is, the Kendall consistence [13] of the above weight vectors

is imperfect

For convenience, we denote the expert weight vector of

set𝐷 as

̃

𝑊∗ = (̃𝑤∗1, ̃𝑤2∗, , ̃𝑤𝑡∗) (13)

Obviously, the relative attribute weights of the set𝐹 could

be got by

𝑊 = (̃𝑤1∗, ̃𝑤∗2, , ̃𝑤𝑡∗)

× (

[𝑎11, 𝑏11] [𝑎12, 𝑏12] ⋅ ⋅ ⋅ [𝑎1𝑛, 𝑏1𝑛]

[𝑎21, 𝑏21] [𝑎22, 𝑏22] ⋅ ⋅ ⋅ [𝑎2𝑛, 𝑏2𝑛]

[𝑎𝑡1, 𝑏𝑡1] [𝑎𝑡2, 𝑏𝑡2] ⋅ ⋅ ⋅ [𝑎𝑡𝑛, 𝑏𝑡𝑛]

The result of the formula [13] is one interval number

column vector; we denote it as

𝑊 = ([𝑎1, 𝑏1], [𝑎2, 𝑏2] , , [𝑎𝑛, 𝑏𝑛])𝑇 (15)

In the following, we would present the new decision

making method

Step 1 Denote the weight vectors which are provided by

experts𝑑𝑖and𝑑𝑗as

𝑊𝑖= (𝑤𝑖1, 𝑤𝑖2, , 𝑤𝑖𝑛) ,

𝑊𝑗= (𝑤𝑗1, 𝑤𝑗2, , 𝑤𝑗𝑛) (16)

The distance between𝑊𝑖and𝑊𝑗would be got by the main

tool of our research, which has been introduced in Section3

The computation follows the formula [5]

Step 2 By formula [5], denote

𝐴1=∑𝑡 𝑘=1󵄨󵄨󵄨󵄨𝑊1− 𝑊𝑘󵄨󵄨󵄨󵄨

=∑𝑡 𝑘=1

√(𝑤𝑘1− 𝑤11)2+ (𝑤𝑘2− 𝑤12)2+ ⋅ ⋅ ⋅ + (𝑤𝑘𝑛− 𝑤1𝑛)2,

𝐴2=∑𝑡 𝑘=1󵄨󵄨󵄨󵄨𝑊2− 𝑊𝑘󵄨󵄨󵄨󵄨

=∑𝑡 𝑘=1

√(𝑤𝑘1− 𝑤21)2+ (𝑤𝑘2− 𝑤22)2+ ⋅ ⋅ ⋅ + (𝑤𝑘𝑛− 𝑤2𝑛)2,

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

𝐴𝑡=∑𝑡 𝑘=1󵄨󵄨󵄨󵄨𝑊𝑡− 𝑊𝑘󵄨󵄨󵄨󵄨

=∑𝑡 𝑘=1

√(𝑤𝑘1− 𝑤𝑡1)2+ (𝑤𝑘2− 𝑤𝑡2)2+ ⋅ ⋅ ⋅ + (𝑤𝑘𝑛− 𝑤𝑡𝑛)2

(17)

Step 3 Take

𝑊∗= (𝐴1

1,𝐴1

2, ,𝐴1

then standardize𝑊∗ We would get the weight vector for experts of set𝐷 Denote 𝑊∗= (𝑤∗1, 𝑤2∗, , 𝑤∗𝑡)𝑇

Step 4 Divide each[𝑎𝑗, 𝑏𝑗] (𝑗 ∈ {1, 2, , 𝑛}) into 𝑛∗ parts The value 𝑛∗ depends on the demand of decision makers Then, we will get a set of segmentation points as

𝑆𝑗= {𝑎𝑗, 𝑎𝑗+𝑛1∗ (𝑏𝑗− 𝑎𝑗) , 𝑎𝑗+𝑛2∗(𝑏𝑗− 𝑎𝑗) , , 𝑏𝑗}

(19)

Step 5 Represent each interval[𝑎𝑗, 𝑏𝑗] (𝑗 ∈ {1, 2, , 𝑛}) by set𝑆𝑗, and represent the vectors𝑊 by one set of real-valued vectors, which would be denoted as ̌𝑊∗here It is easy to see that the element number of the set𝑊 is (𝑛∗)𝑛

Step 6 Take any element𝑊 from ̌𝑊∗, take any𝑖 from 𝑀, and denote

𝑍𝑖= (𝑦𝑖1, 𝑦𝑖2, , 𝑦𝑖𝑛) ⋅ 𝑊 (20) According to comparing each𝑍𝑖(𝑖 ∈ 𝑀), any element 𝑊 from ̌𝑊∗will support one alternative Thus, the set ̌𝑊∗would

be divided into𝑖 subsets We denote the element number of these subsets as𝑛1, 𝑛2, , 𝑛𝑚and denote the support degree for each𝑥𝑖 (𝑖 ∈ 𝑀) as 𝜂𝑖= (𝑛𝑖/𝑛∗)

Step 7 Increase the value𝑛∗gradually and repeat the above steps When the value𝜂𝑖(𝑖 ∈ 𝑀) holds steady to two digits

after the decimal point, the procedure is ended and the final

result𝜂𝑖 (𝑖 ∈ 𝑀) is seen as the support degree for each 𝑥𝑖 (𝑖 ∈

𝑀)

Step 8 By comparing each𝜂𝑖 (𝑖 ∈ 𝑀), we could sort and

select the optimal alternatives

5 An Application Example

In this section, we would present an example to illustrate our

proposed methodology Because the first method is relatively

complex, and the two methods are similar, we only give an

example to verify the first method

This example is coming from the traffic engineering In

this example,𝑋 = {𝑥1, 𝑥2, , 𝑥5} is the set of alternatives,

𝐹 = {𝑓1, 𝑓2, 𝑓3, 𝑓4} is the set of attributes, and all attributes

are of benefit types Consider𝑊 = (𝑤1, 𝑤2, 𝑤3, 𝑤4)𝑇as the

weight vector of all attributes Here, we have no information

about𝑊 The standardized decision matrix is given as

[ [ [ [

0.50 0.80 1.00 0.50 0.70 1.00 0.70 0.90 0.60 0.90 0.60 1.00 0.30 0.90 0.30 0.70 1.00 0.80 0.40 0.80

] ] ] ]

The evaluation vector, which is given towards the set

{𝑥1, 𝑥2, , 𝑥5} and determined by the decision maker, is

𝜆 = (0.45, 0.34, 0.27, 0.25, 0.25) (22)

Now we seek to rank these alternatives and find the most

desirable one

Firstly, according to the method proposed in this paper,

we build one optimal decision model as follows:

min 𝑊∗ = (𝜆∗1− 0.45)2+ (𝜆∗2− 0.34)2+ (𝜆∗3− 0.27)2

+ (𝜆∗4− 0.25)2+ (𝜆∗5− 0.25)2

s.t 𝑤𝑗≥ 0, ∑4

𝑗=1

𝑤𝑗= 1,

(23)

in which

𝜆∗1 = (0.50𝑤1+ 0.80𝑤2+ 1.00𝑤3+ 0.50𝑤4− 0.50)

𝜆∗2 =(0.70𝑤1+ 1.00𝑤2+ 0.70𝑤3+ 0.90𝑤4− 0.70)

𝜆∗3 =(0.60𝑤1+ 0.90𝑤2+ 0.60𝑤3+ 1.00𝑤4− 0.60)

𝜆∗4 =(0.30𝑤1+ 0.90𝑤(1.00 − 0.30)2+ 0.30𝑤3+ 0.70𝑤4− 0.30),

𝜆∗5 =(1.00𝑤1+ 0.80𝑤2+ 0.40𝑤3+ 0.80𝑤4− 0.40)

(24)

Next, by solving the model (23), we would get

𝑊 = (0.0955, 0.0319, 0.5276, 0.3450)𝑇 (25) Afterwards, by using the simple additive weighting method [1], the vector of comprehensive attribute values of each alternative could be obtained and it is

(0.7734, 0.7786, 0.7476, 0.4571, 0.6081)𝑇 (26) Obviously we can conclude that𝑥2would be the optimal alternative and it is inconsistent with the subjective decision information𝜆 Thus, we set the degree of believing for the subjective information as0

Then, we use the method of hypercube segmentation [16]

as a tool to compare the advantages of alternative 𝑥1 and alternative 𝑥2 Results show that 𝑥2 comes to be the best alternative, and its credibility is98.7654%

In a word, there are conflicts between the subjective and objective decision-making information in this case According to our method,𝑥2is the best alternative, with the reliability0 for the subjective information and the reliability 98.7654% for the entire decision

6 Conclusions

Firstly, from the above example, it could be found that the uncertainty of the multisource decision making information has been studied and fused, and the process of the given method is objective, with no subjective factors This is the highlight of our new method

Secondly, though our two methods are all based on the new algorithms of interval numbers, they also have the diversity The first method combines with the optimization theory, and the second method combines with the principle that the minority is subordinate to the majority

Thirdly, from the example it can be seen that it is easy and convenient to use the two new methods, and the numerical example illustrates that our proposed method can deal with the multisource decision-making information well

So, the proposed method may have a higher availability and a better application prospect

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper

Acknowledgments

This work of the first author is supported by the Key Program of the National Natural Science Foundation of China (Grant no 51338003), the National Key Basic Research Program of China (973 Program, Grant no 2012CB725402), and the Science Foundation for Postdoctoral Scientists of Jiangsu Province (Grant no 1301011A) The second author

is supported by the National Natural Science Foundation of China (Grant no 11301474)

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